Fractional reserve banking is a hotly contested term within the libertarian community. Basically, those who are negative about the practice of fractional reserve banking claim it is a cartel-controlled fraud and that bankers who practice fractional reserve banking are criminals who ought to be put in jail and the practice shut down.
Essentially, fractional reserve banking has to do with a bank producing more notes for customers than it has gold in its depository. This was a historical evolution and proponents argue that it is a practice that can be seen in other industries. Airlines provide more tickets to passengers than there are seats. Companies take in fees for machinery that is not yet built and then use just-in-time practices to create the desired equipment.
Proponents of fractional reserve banking also claim it has historical authenticity and that the market ought to decide on the practice, not some higher (presumably statist or communal) authority. This latter was apparently the view of Ludwig von Mises, who believed that in a market setting the populace would eventually reject fractional reserve banking. But he did not want to see it "banned" by government authority; rather, let the free-market decide.
The vitriol about PRIVATE fractional reserve banking in the modern era of free-market thinking may be traced to the famous, idiosyncratic free-market thinker and influential (if polarizing) economist, Murray Rothbard. Rothbard was in some sense a possessive student of the great Austrian economist Ludwig von Mises and adopted his positions, taking them, in some cases, much farther than had Mises. He did this in part because he had a competitive nature and was arguably in competition with Mises's other famous students such as FA Hayek.
Some of Rothbard's elaborations on Mises's points of view were brilliant and Rothbard promulgated an entire theory of anarcho-capitalism merely be extending Mises's thought in logical directions. But when it came to monetary issues, Rothbard exaggerated certain Misesian tendencies and came up with prescriptions that in some cases actually tended to reduce market forces. This can be seen in his focus on "mandated" gold-as-money and also as regards fractional reserve banking.
Both the insistence on gold-as-money and on criminalizing fractional reserve banking have created the current polarized environment within the libertarian community. It is partially personality driven. No one in the Austrian community would argue that central-banking driven fractional reserve banking is a tolerable practice and most would decry it as an outright fraudulent practice. However, when it comes to PRIVATE fractional reserve banking, there is certainly a case to be made, historically and practically, that it is an evolution that may inevitably arise and that it is one the free market ought to deal with – via open and transparent competition between banks – rather than some unnamed authoritarian force that will govern certain bank practices while censoring others. It remains a controversial issue within the libertarian community and one that has not by any means yet been resolved.